
(1) Identification of system status by EGPS states

(a) EGPS state

It is a convex set \(R_s\subseteq \mathcal{D}_{g\phi e}\) with all its elements sharing the same status type, e.g., the state \(s_{11}\) in Fig. 1f, and the probability that the system visits this state (i.e., within \(R_s\)) is bigger than a threshold, i.e., the state is not rare. Empirically, the percentage of a given set of samples falling in \(R_s\) should be larger enough

(b) Prob. EGPS state (dstate vs. cstate )

It is a state that is not rare but prob. (probabilistic) in a sense that each element in \(R_s\) is either Type ‘+’ in a number \(n_S^{+}\) or Type ‘−’ in a number \(n_S^{}\), in two categories:
dstate (Dedicated state): \(max\{n_S^{+},n_S^{}\}\) is significantly bigger than \(min\{n_S^{+},n_S^{}\}\). Empirically, samples falling in \(R_s\) are mostly dedicated to a same status type, e.g., the state \(s_{00}\) in Fig. 1f.
cstate (Confusing state); otherwise, i.e., two status types compete samples in \(R_s\), e.g., the states \(s_{10}\) and \(s_{01}\) in Fig. 1f

(c) cstate (cuttable vs noncuttable)

It is a cstate with at least one convex subset that is able to be cut off as a dstate, e.g., the state \(s_{01}\) in Fig. 1f; otherwise the cstate is said to be noncuttable under the current settings of \(\mathcal{D}_{g\phi e}\), e.g., the state \(s_{10}\) in Fig. 1f

(d) Learning configuration of states

Overall, a set of at least one dstate and cstates (if any) is learned from a given set of samples, featured by not only these states but also their configuration that encodes the locations and mutual relations of these states, as illustrated in Fig. 1h


(2) Refinements of EGPS states

(a) cutting

Cut a cuttable cstate by linear separation, e.g., SVM (Suykens and Vandewalle 1999; Suykens et al. 2002) or FDA by Eqs. (11) and (12) in Ref. Xu (2015a), via refining condition, e.g., the red line cuts \(s_{01}\) in Fig. 2f, which results in one convex subset as a dstate and one sizereduced cstate that may be still cuttable cstate

(b) merging

Merge adjacent dstates if their union is still convex, e.g., merging \(s_2, s_3, s_4\) in Fig. 2f. In addition, merge adjacent cstates, e.g., \(s_5, s_6\) in Fig. 2f

(c) growing

Grow each dstate s with \(Type(s)=\)‘green’ by including those adjacent ’green’ samples if the enlarged subset is still convex, and also grow each dstate s with \(Type(s)=\)‘red’ by including those adjacent ’red’ samples if the enlarged subset is still convex

(d) treating

Use additional conditions (e.g., one more variable is added to \(\mathbf{\phi }\)) such that more ’green’ samples in the cstates become adjacent to and able to be reallocated into some dstates in the above ways


(3) Conditional phenotype analyses based on EGPS states

(a) analysis per dstate

Prognosis analyses test whether \(max\{n_S^{+},n_S^{}\}\) differs from \(min\{n_S^{+},n_S^{}\}\) significantly by \(\chi ^2\) test or Fisher exact test to identify whether this state is good for prognosis, while the boundary of this state indicates the conditions under which the judgement is made. Moreover, prognosis of a unlabelled sample may be made by an oneclass classifier obtained from these conditions


Survival analyses plot KM curves on samples with survival record and make the log rank test or the Cox proportional hazards test


Subtype analyses stratify samples of this state into each subtype, test the enrichment of each subtype in this state, plot KM curves on each stratification, and examine the correlation or the intersection of each subtype to good and bad prognosis, as shown in Fig. 1h

(b) analysis cross dstates

Differentiation test on whether there is a significant difference pairwisely either between samples of different dstates or between samples associated with different values of a phenotype, in one of the following manners:


\(*\) A ttest when we ignore e and merely consider a univariate g;


\(*\) A multivariate test, e.g., Hotelling test Hotelling (1931), BBT test [see Table 6 in Ref. Xu (2015a)], and propertyoriented test [see Algorithm 1 in Ref. Xu (2016)];


\(*\) Modelbased test proposed by Eqs. (29–31) in Ref. Xu (2015a);


\(*\) Logistic or Coxregression. On the lefthand of \(\eta (\phi _t)=\mathbf{b}^T g_t+ \mathbf{a}^T e_t +c+\varepsilon _t\), we test whether one or more of coefficients of \(\mathbf{b}\) are zero and whether one or more of coefficients of \(\mathbf{a}\) (e.g., by the score test or the Wald test) to examine whether the corresponding variables take roles significantly


Staging that is related to subtypes but different, staging involves subtypes in a temporal order. The later stage is usually more serious than the earlier stage, which may be learned via the transfer probabilities \(p(s_is_j)\) cross the states in Fig. 1h


Crossstate integration by comparing the configuration of states to enhance the differentiation study above. Moreover, crossstate combination can further provide better performance, as illustrated in Fig. 1h. Given the output measure \(\zeta _{j,t}\) (e.g., p value, classification error, and predicted regression) for a particular sample t, we may get one weighted average \(\zeta _t=\sum _j \zeta _{j,t} p(s_jt)\), as well as a combined classification rule \(p(+t)=\sum _j p(+s_j) p(s_jt)> p(t)=\sum _j p(s_j) p(s_jt)\)
